A317475 Numbers k such that k^2 | A038199(k).

1, 16, 32, 64, 112, 128, 256, 395, 448, 512, 1024, 1093, 1168, 1368, 1472, 1792, 2013, 2048, 3279, 3344, 3511, 3968, 4096, 5472, 5696, 7168, 7651, 8192, 10533, 14209, 16384, 17488, 19674, 21672, 21888, 22953, 27552, 28672, 31599, 32768, 33883, 34905, 34976

Offset: 1

Author

Amiram Eldar, Jul 29 2018

Keywords

• nonn

Comments

Serret proved in 1855 a generalization of Fermat's little theorem: for b >= 1, Sum_{d|k} mu(d)*b^(k/d) == 0 (mod k). This sequence includes numbers k such that k^2 divides the sum with base b=2.

Includes all the powers of 2 above 8.

An alternative generalization of Wieferich primes (A001220) which are the prime terms of this sequence.

Also numbers k such that k | A059966(k).

Examples

			16 is in the sequence since Sum_{d|16} mu(d)*2^(16/d) = 65280 = 255 * 16^2.
		

References

• Wacław Sierpiński, Elementary Theory of Numbers, Elsevier, North Holland, 1988, page 217.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..250
• Joseph-Alfred Serret, Théorème de Fermat généralisé, Nouvelles Annales de Mathématiques, Vol. 14 (1855), pp. 261-262.

Crossrefs

Cf. A000079, A001220, A038199, A059966, A077816, A182297.

Programs

Mathematica

f[n_] := DivisorSum[n, MoebiusMu[#] * 2^(n/#) &]; Select[Range[1000], Divisible[f[#], #^2] &]

PARI

isok(n) = frac(sumdiv(n, d, moebius(n/d)*(2^d-1))/n^2) == 0; \\ Michel Marcus, Jul 30 2018